# Shape that's getting around

Colin Roberts made his first 'sphericon' as a teenager. Suddenly, it's got a fan club. Victoria Neumark reports.

When I left school I knew I wanted to work with my hands, but I still had this interest in geometry in my system," says Colin Roberts. Mr Roberts, who became a professional joiner and is now 47, invented a new geometrical shape, the sphericon, when he was a teenager. His discovery was finally written up by maths professor Ian Stewart in Scientific American in October 1999.

In his last year at school, 16-year-old Colin Roberts started trying to carve solid versions of the Mobius strip: a twisted ribbon joined into a circle with one continuous surface. Working alone, he found the well-known triangular prism, carving it out of mahogany.

Then he began to wonder whether he could get a one-sided three-dimensional shape without a hole in it. "It suddenly came to me," he says. He conceived the sphericon like the two ends of a bit of wood twisted through 90 degrees. It is, as Professor Stewart puts it, a "cone with a twist", a pleasing shell-like shape. This new solid Mr Roberts made out of wood for his sister, 30 years ago.

"I kept thinking 'its such a simple thing: surely someone else has already discovered it'," he says.

He wrote to toy companies about his discovery, but it seemed "nobody was interested" until in 1997 he saw Ian Stewart on television giving a lecture on symmery. "I wrote to him and he took it up," he says with some surprise.

Fired up once more by Professor Stewart's interest, Mr Roberts has been investigating beyond the single sphericon. Since it has one continuous face, one sphericon can roll round another ad infinitum. Eight can roll round the surface of one. A block of nine can roll round a block of nine, a block of 81 round a block of 81.

A sheet of four can roll around another sheet of four, turning on each other as they roll. They can be joined in six groups of four to map the faces of a truncated octahedron (an octahedron with the points cut off) and then new sets of 24 can join or roll over each other as part of an infinite lattice (the truncated octahedron, along with the cube and the truncated tetrahedron - the three "regular honeycombs" - are the only three shapes which can lattice to infinity and take up all available space).

The maths, according to discussion groups on the Internet, goes on and on, with geodesics and Gauss' Theorem invoked to describe the curvature. And Mr Roberts' enthusiasm has led to speculations on the relation of the sphericon to a shape composed of its inner axes, the "femisphere", on sphericons with different-shaped bases, on sequences between surfaces and edges.

Fifteen-year-old Paul Roberts puts his father's projects onto his computer. He says it's much more interesting than his GCSE maths!

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